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Concept
Banach space
Summary
In mathematics, more specifically in functional analysis, a Banach space (pronounced ˈbanax) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.
Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space".
Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.
A Banach space is a complete normed space
A normed space is a pair
consisting of a vector space over a scalar field (where is commonly or ) together with a distinguished
norm Like all norms, this norm induces a translation invariant
distance function, called the canonical or (norm) induced metric, defined for all vectors by
This makes into a metric space
A sequence is called or or if for every real there exists some index such that
whenever and are greater than
The normed space is called a and the canonical metric is called a if is a , which by definition means for every Cauchy sequence in there exists some such that
where because this sequence's convergence to can equivalently be expressed as:
The norm of a normed space is called a if is a Banach space.
L-semi-inner product
For any normed space there exists an L-semi-inner product on such that for all ; in general, there may be infinitely many L-semi-inner products that satisfy this condition. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces.
Official source
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